Chelsea boss Jose Mourinho values Belgium playmaker Eden Hazard at £200m after reported interest from Real Madrid.
Alan Pardew is weighing up chasing a big-name summer signing to boost Crystal Palace's profile.
The 28-year-old's current contract runs out this summer and, with Polish compatriot Wojciech Szczesny firmly installed as Arsene Wenger's number one, he feels the time is right to leave.
Fabianski is keen to press his international claims, with Szczesny again standing in his way, and he feels the only way he can do that is to be a regular starter at club level.
He has only made four appearances for the Gunners so far this season, two apiece in the Capital One Cup and FA Cup.
"I want to be number one in the club and in the Polish national team. I have learned a lot at London during very positive and challenging periods," Fabianski is quoted as saying in the Daily Star.
"With biggest respect to this great club, I didn't accept the extension offer as I am looking forward to a new challenge in England or abroad."
He may not be the only major departure from the Emirates Stadium this summer with Bacary Sagna also out of contract and reportedly receiving interest from Chelsea and Manchester City as well as a host of European clubs.
Wenger admitted before the 5-1 defeat by Liverpool on Saturday that the right-back could leave for free in the summer, with his desire for a long-term extension and a pay rise making it tough to agree new terms.
"When I say I expect, it's more that I want (him to stay)," said Wenger. "I hope that in the end I am right. We are not close at the moment.
"He wants to keep his options open, because he is at the end of a contract. It is an important contract for him. He is 31 years old. I can understand.
"I'm not alright with that because I would like him to sign just now but I have to accept it. It is his freedom - the decision belongs to him.
"We are always in negotiations with him and we try always to be fair with him. I believe his future is here and any other decision would be wrong."